Pair sequence number
Pair sequence number is the output of a program made by and posted at a googology-related thread of the Japanese site BBS 2ch.net in 2014.First version of BASIC programModified version of BASIC programName of the system and the numberBlog post which introduces pair sequence system, and updated by BashicuHyudora in 2017 on Japanese googology wiki. The algorithm of the program is called the pair sequence system, a weak version of Bashicu matrix system by the same creator. It is verified to terminate, and is supposed to calculate number comparable to \(f_{\psi(\Omega_\omega)+1}(10)\) with respect to Buchholz's function. It is an extension of a system named the primitive sequence system, still by the same author, which generates a number approaching \(f_{\varepsilon_0+1}(10)\).BASIC program of \(\varepsilon_0\) level and a blog post which introduces it A pair sequence is a finite sequence of pairs of nonnegative integers, for example (0,0)(1,1)(2,2)(3,3)(3,2). A pair sequence P works as a function from natural numbers to natural numbers, (though we write Pn rather than P(n)), for example \(n \mapsto (0,0)(1,1)(2,2)(3,3)(3,2)n\) is a function. The function Pn is usually approximated with a function of the form \(H_\alpha\) from the Hardy hierarchy (we note \(P = \alpha\)). For example, \((0,0)(1,1)(2,2)(3,3)(3,2)\) corresponds to \(\psi_0(\Omega_3+\psi_2(\Omega_3+\Omega_2))\) with respect to Buchholz's function. Original BASIC Code In the following program, in the loop starting from "for D=0 to 9" and ending at "next", a number close of \(f_{\psi(\Omega_\omega)}©\) with respect to Buchholz's function is generated. The program repeats this loop 10 times and finally outputs a number close of \(f_{\psi(\Omega_\omega)+1}(10)\) with respect to Buchholz's function. dim A∞,B∞:C=9 for D=0 to 9 for E=0 to C AE=E:BE=E next for F=C to 0 step -1 C=C*C for G=0 to C if AF=0 | (AF-GBashicu matrix calculator shows the calculation process of pair sequence system. BM1 corresponds to the original version. Here are some examples of the calculation of some pair sequences. The algorithm is modified so that it always take n=2. * (0,0)(1,1) * (0,0)(1,1)(1,1) * (0,0)(1,1)(2,0) * (0,0)(1,1)(2,1) * (0,0)(1,1)(2,2) * (0,0)(1,1)(2,2)(3,3) * (0,0)(1,1)(2,2)(3,3)(4,4) Corresponding ordinals It is verified that each standard pair sequence \(M\) corresponds to an ordinal \(\textrm{Trans}(M)\) below \(\psi(\Omega_{\omega})\) so that the expansion expansion of \(M\) gives a strictly increasing sequence of ordinals below \(\textrm{Trans}(M)\). In particular, it implies that pair sequence system restricted to standard pair sequences gives a total computable function whose stractural well-ordering is of ordinal type bounded by \(\psi(\Omega_{\omega})\). It is also strongly believed in this community that the structural well-ordering is of ordinal type \(\psi(\Omega_{\omega})\). The following are analyses of the structural well-ordering based on an unspecified OCF which is different from Buchholz's function without a proof: Up to \(\varepsilon_0\) When all the values of the second row are 0, it is the same as the primitive sequence system. We have: \begin{array}{ll} (0,0) &=& 1 \\ (0,0)(0,0) &=& 2 \\ (0,0)(0,0)(0,0) &=& 3 \\ (0,0)(1,0) &=& \omega \\ (0,0)(1,0)(0,0)(0,0) &=& \omega+2 \\ (0,0)(1,0)(0,0)(1,0) &=& \omega \cdot 2 \\ (0,0)(1,0)(1,0) &=& \omega^2 \\ (0,0)(1,0)(1,0)(0,0)(1,0) &=& \omega^2+\omega \\ (0,0)(1,0)(2,0) &=& \omega^\omega \\ (0,0)(1,0)(2,0)(3,0) &=& \omega^{\omega^\omega} \\ (0,0)(1,0)(2,0)(3,0)(4,0) &=& \omega^{\omega^{\omega^\omega}} \\ \end{array} (0,0)(1,1) has fundamental sequence as follows. Here, n is not changed. \begin{array}{ll} (0,0)(1,1)1 &=& (0,0)(1,0)1 \\ (0,0)(1,1)2 &=& (0,0)(1,0)(2,0)2 \\ (0,0)(1,1)3 &=& (0,0)(1,0)(2,0)(3,0)3 \\ (0,0)(1,1)4 &=& (0,0)(1,0)(2,0)(3,0)(4,0)4 \\ \end{array} Therefore, \(\{\omega, \omega^\omega, \omega^{\omega^\omega}, \omega^{\omega^{\omega^\omega}}, \ldots\}\) and \begin{array}{ll} (0,0)(1,1) &=& \varepsilon_0 \\ \end{array} Up to \(\varepsilon_1\) As for (0,0)(1,1)(1,0), \[(0,0)(1,1)(1,0)4 = (0,0)(1,1)(0,0)(1,1)(0,0)(1,1)(0,0)(1,1)(0,0)(1,1)4\] and the fundamental sequence is \begin{array}{ll} (0,0)(1,1) &=& \varepsilon_0 \\ (0,0)(1,1)(0,0)(1,1) &=& \varepsilon_0 \cdot 2 \\ (0,0)(1,1)(0,0)(1,1)(0,0)(1,1) &=& \varepsilon_0 \cdot 3 \\ (0,0)(1,1)(0,0)(1,1)(0,0)(1,1)(0,0)(1,1) &=& \varepsilon_0 \cdot 4 \\ (0,0)(1,1)(0,0)(1,1)(0,0)(1,1)(0,0)(1,1)(0,0)(1,1) &=& \varepsilon_0 \cdot 5 \\ \end{array} Therefore, \= \varepsilon_0 \cdot \omega\ \[(0,0)(1,1)(1,0)(1,0)2 = (0,0)(1,1)(1,0)(0,0)(1,1)(1,0)(0,0)(1,1)(1,0)2\] has fundamental sequence of \begin{array}{ll} (0,0)(1,1)(1,0) &=& \varepsilon_0 \cdot \omega \\ (0,0)(1,1)(1,0)(0,0)(1,1)(1,0) &=& \varepsilon_0 \cdot \omega \cdot 2 \\ (0,0)(1,1)(1,0)(0,0)(1,1)(1,0)(0,0)(1,1)(1,0) &=& \varepsilon_0 \cdot \omega \cdot 3 \\ \end{array} Therefore, \= \varepsilon_0 \cdot \omega^2 \ In this way, adding (1,0) to the end of the sequence makes the ordinal \(\omega\) times. Adding (1,0)(2,0) to the end of the sequence \[(0,0)(1,1)(1,0)(2,0)4 = (0,0)(1,1)(1,0)(1,0)(1,0)(1,0)(1,0)4\] corresponds to multiplying \(\omega^\omega\) to the ordinal, and therefore \= \varepsilon_0 \cdot \omega^\omega\ As for (0,0)(1,1)(1,1), \[(0,0)(1,1)(1,1)4 = (0,0)(1,1)(1,0)(2,1)(2,0)(3,1)(3,0)(4,1)(4,0)(5,1)4\] and the following fundamental sequence is obtained. \begin{array}{ll} (0,0)(1,1) &=& \varepsilon_0 \\ (0,0)(1,1)(1,0)(2,1) &=& \varepsilon_0^2 \\ (0,0)(1,1)(1,0)(2,1)(2,0)(3,1) &=& \varepsilon_0^{\varepsilon_0} \\ (0,0)(1,1)(1,0)(2,1)(2,0)(3,1)(3,0)(4,1) &=& \varepsilon_0^{\varepsilon_0^2} \\ (0,0)(1,1)(1,0)(2,1)(2,0)(3,1)(3,0)(4,1)(4,0)(5,1) &=& \varepsilon_0^{\varepsilon_0^{\varepsilon_0}} \\ \end{array} Therefore, \= \varepsilon_1 = \psi(1) \ Up to Feferman–Schütte ordinal = \(\Gamma_0\) Similar calculation results in: \begin{eqnarray*} (0,0)(1,1)(2,0) &=& \varepsilon_{\omega} = \psi(\omega) \\ (0,0)(1,1)(2,0)(2,0) &=& \varepsilon_{\omega^2} = \psi(\omega^2) \\ (0,0)(1,1)(2,0)(3,0) &=& \varepsilon_{\omega^\omega} = \psi(\omega^\omega) \\ (0,0)(1,1)(2,0)(3,1) &=& \varepsilon_{\varepsilon_0} = \psi(\psi(0)) \\ (0,0)(1,1)(2,0)(3,1)(4,0)(5,1) &=& \varepsilon_{\varepsilon_{\varepsilon_0}} = \psi(\psi(\psi(0))) \\ (0,0)(1,1)(2,1) &=& \zeta_0 = \varphi(2,0) = \psi(\Omega) \\ (0,0)(1,1)(2,1)(1,1) &=& \varepsilon_{\zeta_0+1} \\ (0,0)(1,1)(2,1)(1,1)(2,1) &=& \zeta_1= \varphi(2,1) \\ (0,0)(1,1)(2,1)(2,0) &=& \zeta_\omega = \varphi(2,\omega) \\ (0,0)(1,1)(2,1)(2,1) &=& \eta_0= \varphi(3,0) \\ (0,0)(1,1)(2,1)(2,1)(2,1) &=& \varphi(4,0) \\ (0,0)(1,1)(2,1)(3,0) &=& \varphi(\omega,0) \\ (0,0)(1,1)(2,1)(3,1) &=& \Gamma_0 = \varphi(1,0,0) = \psi(\Omega^\Omega) \end{eqnarray*} Up to Large Veblen ordinal = \(\psi(\Omega^{\Omega^\Omega})\) \begin{eqnarray*} (0,0)(1,1)(2,1)(3,1)(1,1) &=& \varepsilon_{\Gamma_0+1} \\ (0,0)(1,1)(2,1)(3,1)(1,1)(2,1) &=& \zeta_{\Gamma_0+1} \\ (0,0)(1,1)(2,1)(3,1)(1,1)(2,1)(3,1) &=& \Gamma_1 = \varphi(1,0,1) \\ (0,0)(1,1)(2,1)(3,1)(2,0) &=& \Gamma_\omega = \varphi(1,0,\omega) \\ (0,0)(1,1)(2,1)(3,1)(2,1) &=& \varphi(1,1,0) \\ (0,0)(1,1)(2,1)(3,1)(2,1)(1,1)(2,1)(3,1)(2,1) &=& \varphi(1,1,1) \\ (0,0)(1,1)(2,1)(3,1)(2,1)(2,0) &=& \varphi(1,1,\omega) \\ (0,0)(1,1)(2,1)(3,1)(2,1)(2,1) &=& \varphi(1,2,0) \\ (0,0)(1,1)(2,1)(3,1)(2,1)(3,1) &=& \varphi(2,0,0) \\ (0,0)(1,1)(2,1)(3,1)(3,0) &=& \varphi(\omega,0,0) \\ (0,0)(1,1)(2,1)(3,1)(3,1) &=& \varphi(1,0,0,0) \\ (0,0)(1,1)(2,1)(3,1)(3,1)(1,1)(2,1)(3,1)(3,1) &=& \varphi(1,0,0,1) \\ (0,0)(1,1)(2,1)(3,1)(3,1)(2,1) &=& \varphi(1,0,1,0) \\ (0,0)(1,1)(2,1)(3,1)(3,1)(2,1)(3,1) &=& \varphi(1,1,0,0) \\ (0,0)(1,1)(2,1)(3,1)(3,1)(2,1)(3,1)(2,1)(3,1) &=& \varphi(1,2,0,0) \\ (0,0)(1,1)(2,1)(3,1)(3,1)(2,1)(3,1)(3,1) &=& \varphi(2,0,0,0) \\ (0,0)(1,1)(2,1)(3,1)(3,1)(2,1)(3,1)(3,1)(2,1)(3,1)(3,1) &=& \varphi(3,0,0,0) \\ (0,0)(1,1)(2,1)(3,1)(3,1)(3,0) &=& \varphi(\omega,0,0,0) \\ (0,0)(1,1)(2,1)(3,1)(3,1)(3,1) &=& \varphi(1,0,0,0,0) = \psi(\Omega^{\Omega^3}) \\ (0,0)(1,1)(2,1)(3,1)(3,1)(3,1)(3,1) &=& \varphi(1,0,0,0,0,0) = \psi(\Omega^{\Omega^4}) \\ (0,0)(1,1)(2,1)(3,1)(4,0)&=& \psi(\Omega^{\Omega^\omega}) \text{(SVO)} \\ (0,0)(1,1)(2,1)(3,1)(4,0)(3,1) &=& \psi(\Omega^{\Omega^{\omega+1}}) \\ (0,0)(1,1)(2,1)(3,1)(4,0)(4,0) &=& \psi(\Omega^{\Omega^{\omega^2}}) \\ (0,0)(1,1)(2,1)(3,1)(4,0)(5,0) &=& \psi(\Omega^{\Omega^{\omega^\omega}}) \\ (0,0)(1,1)(2,1)(3,1)(4,0)(5,1) &=& \psi(\Omega^{\Omega^{\varepsilon_0}}) \\ (0,0)(1,1)(2,1)(3,1)(4,1) &=& \psi(\Omega^{\Omega^\Omega}) \text{(LVO)} \end{eqnarray*} Up to Bachmann-Howard ordinal \begin{eqnarray*} (0,0)(1,1)(2,1)(3,1)(4,1)(4,0) &=& \psi(\Omega^{\Omega^{\Omega \cdot \omega}}) \\ (0,0)(1,1)(2,1)(3,1)(4,1)(4,1) &=& \psi(\Omega^{\Omega^{\Omega^2}}) \\ (0,0)(1,1)(2,1)(3,1)(4,1)(5,0) &=& \psi(\Omega^{\Omega^{\Omega^\omega}}) \\ (0,0)(1,1)(2,1)(3,1)(4,1)(5,1) &=& \psi(\Omega^{\Omega^{\Omega^\Omega}}) \\ (0,0)(1,1)(2,1)(3,1)(4,1)(5,1)(6,1) &=& ψ(\Omega^{\Omega^{\Omega^{\Omega^\Omega}}}) \\ (0,0)(1,1)(2,1)(3,1)(4,1)(5,1)(6,1)(7,1) &=& ψ(\Omega^{\Omega^{\Omega^{\Omega^{\Omega^\Omega}}}}) \\ (0,0)(1,1)(2,2) &=& \psi(\varepsilon_{\Omega+1}) = \psi(\psi_1(0)) \end{eqnarray*} Up to \(\psi(\Omega_\omega)\) \begin{eqnarray*} (0,0)(1,1)(2,2)(0,0) &=& \psi(\psi_1(0))+1 \\ (0,0)(1,1)(2,2)(1,0) &=& \psi(\psi_1(0)) \omega \\ (0,0)(1,1)(2,2)(2,0) &=& \psi(\psi_1(0) \omega) \\ (0,0)(1,1)(2,2)(3,0) &=& \psi(\psi_1(\omega)) \\ (0,0)(1,1)(2,2)(3,0)(4,0) &=& \psi(\psi_1(\omega^\omega)) \\ (0,0)(1,1)(2,2)(3,0)(4,1) &=& \psi(\psi_1(\psi(0)))=\psi(\psi_1(\varepsilon_0)) \\ (0,0)(1,1)(2,2)(3,1) &=& \psi(\psi_1(\Omega)) \\ (0,0)(1,1)(2,2)(3,2) &=& \psi(\psi_1(\Omega_2)) \\ (0,0)(1,1)(2,2)(3,3) &=& \psi(\psi_1(\psi_2(0))) \\ (0,0)(1,1)(2,2)(3,3)(4,4) &=& \psi(\psi_1(\psi_2(\psi_3(0)))) \\ (0,0)(1,1)(2,2)(3,3)...(9,9) &=& \psi(\psi_1(\psi_2(\psi_3(\psi_4(\psi_5(\psi_6(\psi_7(\psi_8(0))))))))) \end{eqnarray*} By defining \(Pair(n) = (0,0)(1,1) \ldots (n,n)n\), one has an expectation \\approx f_{\psi(\Omega_\omega)}(n)\ with respect to Buchholz's function and the canonical system of fundamental sequences. Be careful that the \(\psi\) in the analyses is not Buchholz's function. Sources See also ja:ペア数列数zh:對數列系統 Category:Bashicu matrix system Category:Higher computable level Category:Googology in Asia Category:Numbers